Regularity index of fat points in the projective plane
The aim of this paper is to find a bound for the minimum value of \(t > 0\) for which a given scheme of fat points in \(\mathbb{P}^ 2\) imposes independent conditions on curves of degree \(t\). A scheme \(X\) of ``fat points'' is a 0-dimensional scheme defined by a homogeneous ideal \(I = {\mathfrak p}_ 1^{m_ 1} \cap {\mathfrak p}_ 2^{m_ 2} \cap \cdots \cap {\mathfrak p}_ s^{m_ s} \subseteq k [x_ 0,x_ 1,x_ 2]\), where \(k\) is an algebraically closed field with \(\text{char} k = 0\). Every vector space \(I_ t\) represents the curves of degree \(t\) having multiplicity at least \(m_ i\) at each \(P_ i\), \(i = 1,\dots, s\). Let \(\tau = \tau (I)\) be the index of regularity of \(I\) (or of \(X)\), i.e. the minimum value of \(t\) for which \(X\) imposes independent conditions on curves of degree \(t\) (equivalently, the minimum \(t\) for which \(H^ 1 (\mathbb{P}^ 2, {\mathcal I} (t)) = 0\), where \({\mathcal I}\) is the ideal sheaf of \(X\), or also the minimum \(t\) for which \(H(X,t) = \deg X = \sum {m_ i + 1 \choose 2}\), where \(H(X,.)\) is the Hilbert function of \(X)\). -- The most general upper bound possible for \(\tau\) is: \(\tau \leq \sum m_ i - 1\), which is attained when the \(s\) points are collinear. If we suppose that the points are in general position (no three on a line) and that \(m_ 1 \leq \cdots \leq m_ s\), then a better bound has been given by \textit{B. Segre}: \(\tau \leq \max \{m_ 1 + m_ 2 - 1\), \([\sum m_ i/2]\}\). In this paper a bound is given that generalizes the two mentioned above; for any scheme \(X\) of fat points, we have \(\tau \leq \max \{h - 1, [\sum m_ i/2]\}\), where \(h\) is the maximum among the sums of multiplicities taken over all subsets of collinear points. Moreover, if \(\max\{h - 1, [\sum m_ i/2]\} = h - 1\), then \(\tau = h - 1\).
scientific article; zbMATH DE number 705010